Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 12 / vi / Solution

Reversal has one inversion of a permutation for every pair of indices, hence $\left(\genfrac{}{}{0ex}{}{n}{2}\right)=n(n-1)/2$ inversions of a permutation. The sign of a permutation givesIt is $+1$ for $n\equiv 0,1(\mathrm{mod}4)$ and $-1$ for $n\equiv 2,3(\mathrm{mod}4)$.